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Abstract

We study the following graph partitioning problem: Given two positive integers C
and Δ, find the least integer M(C,Δ) such that the edges of any graph with maximum degree at
most Δ can be partitioned into subgraphs with at most C edges and each vertex appears in at most
M(C,Δ) subgraphs. This problem is naturally motivated by traffic grooming, which is a major
issue in optical networks. Namely, we introduce a new pseudodynamic model of traffic grooming in
unidirectional rings, in which the aim is to design a network able to support any request graph with
a given bounded degree. We show that optimizing the equipment cost under this model is essentially
equivalent to determining the parameter M(C, Δ). We establish the value of M(C, Δ) for almost all
values of C and Δ, leaving open only the case where Δ ≥ 5 is odd, Δ (mod 2C) is between 3 and
C − 1, C ≥ 4, and the request graph does not contain a perfect matching. For these open cases, we
provide upper bounds that differ from the optimal value by at most one.